arXiv subjects

Anatolij Dvurečenskij

Publications and source records attributed to Anatolij Dvurečenskij.

42 records · Page 3Linked to original sources

arXiv2012

Smearing of Observables and Spectral Measures on Quantum Structures

An observable on a quantum structure is any $σ$-homomorphism of quantum structures from the Borel $σ$-algebra of the real line into the quantum structure which is in our case a monotone $σ$-complete effect algebras with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean $σ$-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there is its spectral measure.

math-ph↗

arXiv2011

On Normal-Valued Basic Pseudo Hoops

We show that every pseudo hoop satisfies the Riesz Decomposition Property. We visualize basic pseudo hoops by functions on a linearly ordered set. Finally, we study normal-valued basic pseudo hoops giving a countable base of equations for them.

math.AC↗

arXiv2010

Representation of States on Effect-Tribes and Effect Algebras by Integrals

We describe $σ$-additive states on effect-tribes by integrals. Effect-tribes are monotone $σ$-complete effect algebras of functions where operations are defined by points. Then we show that every state on an effect algebra is an integral through a Borel regular probability measure. Finally, we show that every $σ$-convex combination of extremal states on a monotone $σ$-complete effect algebra is a Jauch-Piron state.

math.FA↗

arXiv2010

Every State on Interval Effect Algebra is Integral

We show that every state on an interval effect algebra is an integral through some regular Borel probability measure defined on the Borel $σ$-algebra of a compact Hausdorff simplex. This is true for every effect algebra satisfying (RDP) or for every MV-algebra. In addition, we show that each state on an effect subalgebra of an interval effect algebra $E$ can be extended to a state on $E.$ Our method represents also every state on the set of effect operators of a Hilbert space as an integral

math.FA↗

arXiv2009

State BL-algebras

The concept of a state MV-algebra was firstly introduced by Flaminio and Montagna in \cite{FlMo0} and \cite{FlMo} as an MV-algebra with internal state as a unary operation. Di Nola and Dvurečenskij gave a stronger version of a state MV-algebra in \cite{DiDV1}, \cite{DiDV2}. In the present paper we introduce the notion of a state BL-algebra, or more precisely, a BL-algebra with internal state. We present different types of state BL-algebras, like strong state BL-algebras and state-morphism BL-algebras, and we study some classes of state BL-algebras. In addition, we give a sample of important examples of state BL-algebras and present some open problems.

math.LO↗

arXiv2009

Measures, states and de Finetti maps on pseudo-BCK algebras

In this paper, we extend the notions of states and measures presented in \cite{DvPu} to the case of pseudo-BCK algebras and study similar properties. We prove that, under some conditions, the notion of a state in the sense of \cite{DvPu} coincides with the Bosbach state, and we extend to the case of pseudo-BCK algebras some results proved by J. Kühr only for pseudo-BCK semilattices. We characterize extremal states, and show that the quotient pseudo-BCK algebra over the kernel of a measure can be embedded into the negative cone of an archimedean $\ell$-group. Additionally, we introduce a Borel state and using results by J. Kühr and D. Mundici from \cite{Kumu}, we prove a relationship between de Finetti maps, Bosbach states and Borel states.

math.LO↗